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On fractional integration formulae for Aleph functions. (English) Zbl 1242.33021
Summary: This paper is devoted to the study of a new special function, which is called Aleph function, according to the symbol used to represent this function. This function is an extension of the $$I$$-function, which itself is a generalization of the well-known and familiar $$G$$- and $$H$$-functions in one variable. In this paper, a notation and a complete definition of the Aleph function will be presented. Fractional integration of Aleph functions, in which the argument of the Aleph function contains a factor $$t^{\lambda }(1 - t)^{\mu }, \lambda , \mu > 0$$, will be investigated. The results derived are of most general character and include many results given earlier by various authors including A. A. Kilbas [Fract. Calc. Appl. Anal. 8, No. 2, 113–126 (2005; Zbl 1144.26008)], A. A. Kilbas and M. Saigo [Fukuoka Univ. Sci. Rep. 28, No. 2, 41–51 (1998; Zbl 0923.33006)] and L. Galué [Int. J. Appl. Math. 10, No. 3, 255–267 (2002; Zbl 1036.33002)] and others. The results obtained form the key formulae for the results on various potentially useful special functions of physical and biological sciences and technology available in the literature.

##### MSC:
 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions) 26A33 Fractional derivatives and integrals
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